given f(x)=-2x+3 and g(x)=1/2x-2 find x when f(g(x))=9

To find the value of \( x \) when \( f(g(x)) = 9 \), follow these steps:

### Given:
\[
f(x) = -2x + 3
\]
\[
g(x) = \frac{1}{2}x - 2
\]

### Step 1: Find \( f(g(x)) \)
Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f\left(\frac{1}{2}x - 2\right)
\]
Now, replace \( x \) in \( f(x) = -2x + 3 \) with \( \frac{1}{2}x - 2 \):
\[
f(g(x)) = -2\left(\frac{1}{2}x - 2\right) + 3
\]
Distribute the \(-2\):
\[
f(g(x)) = -2 \cdot \frac{1}{2}x + 4 + 3
\]
Simplify:
\[
f(g(x)) = -x + 7
\]

### Step 2: Set \( f(g(x)) = 9 \)
\[
-x + 7 = 9
\]

### Step 3: Solve for \( x \)
Subtract 7 from both sides:
\[
-x = 2
\]
Multiply by \(-1\) to solve for \( x \):
\[
x = -2
\]

So, \( x = -2 \).

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:
1. What is \( g(f(x)) \) for the given functions?
2. How would the solution change if \( f(g(x)) = -9 \)?
3. What is the inverse of \( f(x) \) and \( g(x) \)?
4. How do the graphs of \( f(x) \) and \( g(x) \) look?
5. Can you find \( f(x) = 0 \) for the given \( f(x) \)?

**Tip:** When solving equations involving function compositions, always carefully apply the functions step by step to avoid mistakes.

Learn how to solve the equation f(g(x)) = 9 using function composition. Step-by-step solution provided for the functions f(x) = -2x + 3 and g(x) = 1/2x - 2. Suitable for Grades 9-12 mathematics.

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